Math 1410 – Test #3B – Part 1
Spring 2010
Score: _____
Name__________________________________ Row_____
25
Calculators are not allowed on this page of the test. When you are done with this page,
exchange it for the rest of the test.
Q1: Evaluate the integrals.
A. Evaluate the definite integral.
2
B. Find the antiderivative
5 4
+ 𝑡 𝑑𝑡
𝑡
2𝑠 2 − 3 𝑑𝑠
1
C. Use a 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 to find the antiderivative
2𝑥 2
𝑑𝑥
𝑥3 + 5 4
D. Circle all correct answers.
Which of the following can be converted to the form
1
𝑑𝑤 using a 𝑤 − 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛?
𝑤
𝐴.
𝐵.
𝑥2
2
𝑑𝑥
+4
3𝑥
𝑑𝑥
+4
𝑥2
𝐶.
2𝑥 2
𝑑𝑥
𝑥2 + 4
𝐷.
cos 𝑥
𝑑𝑥
sin 𝑥 + 2
𝐸.
2𝑥
𝑥2 + 1
𝑑𝑥
Math 1410 – Test #3A-2
Spring 2010
Score: _____
80
#_____ Name__________________________________
Show all work. Label answers to word problems.
Q1: A curve representing the total number of people, 𝑃, infected with a virus has the shape of a logistic curve
𝑃=
𝐿
1 + 𝐶𝑒 −𝑘𝑡
with time 𝑡 in weeks. Suppose that initially 10 people are infected with the virus and that in the early weeks the
number of people infected is increasing approximately exponentially, with a continuous growth rate of 1.8. It is
estimated that, in the long run, approximately 6000 people will become infected.
A. Find the values of the constants 𝐿, 𝐶, 𝑘 and rewrite the equation.
10
B. At what time will the rate at which people are infected begin to decrease? How many people are
infected at this time?
Q2: Roger is practicing for a marathon and his friend,
pedaling behind him, records his speed, in miles per hour,
every 15 minutes during his 90 minute run. Assuming that
Roger's speed never was increasing, find upper and
lower estimates for the distance that he ran.
Time since start
in minutes
0
Speed in
miles per hour
12 11 10 10
15 30 45 60 75 90
8
7
0
5
Page 2 of 3
Q3:
1. Place the integrals in order of their values, from smallest to largest.
Refer to each integral by its letter 𝐴 − 𝐹.
𝑏
𝐴.
𝑐
𝑓 𝑥 𝑑𝑥
𝐵.
𝑎
𝑐
𝑓 𝑥 𝑑𝑥
𝐶.
𝑓 𝑥 𝑑𝑥
𝑎
𝑑
𝑓 𝑥 𝑑𝑥
𝑏
smallest _____
𝑓(𝑥)
𝑑
𝑎
𝐷.
10
𝐸.
𝑑
𝑓 𝑥 𝑑𝑥
𝐹.
𝑏
_____
𝑓 𝑥 𝑑𝑥
a
𝑐
_____
_____
_____
b
c
d
_____ largest
2. Using good notation, write the integral expression that will find the area of the region bounded by 𝑦 = 𝑓(𝑥)
and the 𝑥 − 𝑎𝑥𝑖𝑠 between 𝑥 = 𝑎 and 𝑥 = 𝑑.
𝑎𝑟𝑒𝑎 =
Q4: After a foreign substance is introduced into the blood, the rate at which antibodies are made is given by
𝑟 𝑡 =
𝑡2
𝑡
thousands of antibodies per minute
+1
where time, 𝑡, is in minutes. Assuming there are no antibodies present at 𝑡 = 0, find the total quantity of
antibodies in the blood at the end of 4 minutes. Set up the calculation & evaluate. A calculator answer is fine.
10
Page 3 of 3
Q5: Shown is the graph of 𝐹′(𝑡). Using the fact that 𝐹(0) = 5 , sketch the graph of 𝐹 𝑡 . Label the values of the
known points.
𝐹(𝑡)
′
𝐹 (𝑡)
10
Area = 6
Area =2
2
4
6
2
8
4
6
8
Area = 12
Q6: Set up the definite integral to find the area bounded by
y
𝑦 = 5 − 𝑥 2 and 𝑦 = −4 . Notation counts.
Set up only – do not evaluate.
6
x
Q7: A potato is cooking in an oven and its temperature, in degrees Fahrenheit, is given by 𝐹(𝑡) , where 𝑡 is minutes
since it was placed in the oven. Explain in words the meaning of
1
25
25
𝐹 𝑡 𝑑𝑡 = 100
0
4